Weak mutually unbiased bases

TL;DR

This paper introduces weak mutually unbiased bases in quantum systems over finite groups, exploring their properties, geometric duality, and relation to standard mutually unbiased bases, especially for prime dimensions.

Contribution

It defines and analyzes weak mutually unbiased bases, revealing their geometric duality with phase space lines and their equivalence to MUBs in prime dimensions.

Findings

Weak mutually unbiased bases are characterized by specific overlap conditions.

A duality exists between these bases and maximal lines in phase space.

In prime dimensions, they coincide with standard mutually unbiased bases.

Abstract

Quantum systems with variables in ${\mathbb Z}(d)$ are considered. The properties of lines in the ${\mathbb Z}(d)\times {\mathbb Z}(d)$ phase space of these systems, are studied. Weak mutually unbiased bases in these systems are defined as bases for which the overlap of any two vectors in two different bases, is equal to $d^{-1/2}$ or alternatively to one of the $d_i^{-1/2},0$ (where $d_i$ is a divisor of $d$ apart from $d,1$). They are designed for the geometry of the ${\mathbb Z}(d)\times {\mathbb Z}(d)$ phase space, in the sense that there is a duality between the weak mutually unbiased bases and the maximal lines through the origin. In the special case of prime $d$, there are no divisors of $d$ apart from $1,d$ and the weak mutually unbiased bases are mutually unbiased bases.